Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential
We prove that a neutral atom in mean-field approximation has O(4) symmetry and this fact explains the empirical [n+l,n]-rule or Madelung rule which describes effectively periods, structure and other properties of the Mendeleev table of chemical elements.
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irk-123456789-1486402019-02-19T01:31:48Z Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential Belokolos, E.D. We prove that a neutral atom in mean-field approximation has O(4) symmetry and this fact explains the empirical [n+l,n]-rule or Madelung rule which describes effectively periods, structure and other properties of the Mendeleev table of chemical elements. 2017 Article Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential / E.D. Belokolos // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81Q05; 81V45 DOI:10.3842/SIGMA.2017.038 http://dspace.nbuv.gov.ua/handle/123456789/148640 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We prove that a neutral atom in mean-field approximation has O(4) symmetry and this fact explains the empirical [n+l,n]-rule or Madelung rule which describes effectively periods, structure and other properties of the Mendeleev table of chemical elements. |
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Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential |
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Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential |
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mendeleev table: a proof of madelung rule and atomic tietz potential |
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Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential / E.D. Belokolos // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 34 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 038, 15 pages
Mendeleev Table: a Proof of Madelung Rule
and Atomic Tietz Potential
Eugene D. BELOKOLOS
Department of Theoretical Physics, Institute of Magnetism,
National Academy of Sciences of Ukraine, 36-b Vernadsky Blvd., Kyiv, 252142, Ukraine
E-mail: bel@imag.kiev.ua
Received February 27, 2017, in final form May 22, 2017; Published online June 07, 2017
https://doi.org/10.3842/SIGMA.2017.038
Abstract. We prove that a neutral atom in mean-field approximation has O(4) symmetry
and this fact explains the empirical [n+l, n]-rule or Madelung rule which describes effectively
periods, structure and other properties of the Mendeleev table of chemical elements.
Key words: Madelung rule; Mendeleev periodic system of elements; Tietz potential
2010 Mathematics Subject Classification: 81Q05; 81V45
1 Introduction
In 1869 D.I. Mendeleev discovered a periodic dependence of chemical element properties on Z
with periods
2, 8, 8, 18, 18, 32, . . . , (1.1)
where Z is a number of electrons in the chemical element atom1.
With creation of quantum mechanics physicists tried to explain the Mendeleev empirical law
in terms of the one-particle quantum numbers
n, l, m, σ,
where n = nr+l+1 is a principal quantum number, nr is a radial quantum number, l is an orbital
quantum number, m is a magnetic quantum number, −l ≤ m ≤ l, σ = ±1/2 is a projection
of electron spin. For example, energy levels of the Hydrogen atom, which has O(4) symmetry,
depend on the principal quantum number n only and are degenerate in other quantum numbers:
l, 0 ≤ l ≤ n − 1 (so called “accidental” degeneracy); m, −l ≤ m ≤ l; σ, σ = ±1/2. It is
a so called [n, l]-rule. According to it the energy spectrum of atom consists of electron shells,
enumerated by the principal quantum number n and having the degeneracy
Nn =
n∑
l=0
2(2l + 1) = 2n2.
This formula gives the following set of periods
2, 8, 18, 32, . . . . (1.2)
Comparing the period sequences (1.1) and (1.2), we can explain only the first two periods of
the Mendeleev table. Although we see a remarkable fact that (1) the period lengths have cardi-
nalities that correspond to the Hydrogen degeneracy dimensions, and (2) the same cardinalities
always occur in pairs, except for the very first one. In this paper we show that these similarities
are not accidental.
1See https://en.wikipedia.org/wiki/Periodic_table.
mailto:bel@imag.kiev.ua
https://doi.org/10.3842/SIGMA.2017.038
https://en.wikipedia.org/wiki/Periodic_table
2 E.D. Belokolos
2 The Madelung [n + l, n]-rule
We get an exact expression for the Mendeleev periods and other properties of the Mendeleev
table with the Madelung [n+ l, n]-rule [22]).
The [n + l, n]-rule asserts: with growth of atomic charge Z the electrons fill up in atom
consecutively the one-particle states with the least possible value of the quantum number n+ l;
and, for a given value n + l, the electrons fill up states with the least possible value of the
quantum number n.
Here it is reasonable to introduce the quantum number
M = n+ l = nr + 2l + 1.
The [n+l, n]-rule is in fact an algorithm for consecutive building-up of atoms2. It describes the
states for each of about 5000 electrons of the atoms to corresponding elements of the periodic
system (indeed, now we have 118 chemical elements, and for approximately 100 elements we
know every state of electron configuration, that is
100∑
1
Z = 5050 states) and predicts correctly the
real electron configurations of all elements of periodic system with small number of exclusions.
There exist only 19 elements (Cr, Cu, Nb, Mo, Ru, Rh, Pd, Ag, La, Ce, Gd, Pt, Au, Ac, Th,
Pa, U, Np, Cm), whose electron configurations differ from the configurations predicted with
the [n+ l, n]-rule [24].
The [n+ l, n]-rule allows to obtain relations between the order number Z of chemical element
and quantum numbers M , n, l of the appropriate electron configuration.
Let us designate the order number Z of chemical element in which electrons from the (n+ l)-
subgroups, (n, l)-subgroups, n-subgroups, l-subgroups or nr-subgroups appear for the first time
by symbols
Zn+l, Zn,l, Zn, Zl, Znr .
Let us designate an order number Z of a chemical element in which the (n+ l)-subgroups, (n, l)-
subgroups, n-subgroups, l-subgroups or nr-subgroups are filled up completely for the first time
by symbols
n+lZ, n,lZ, nZ, lZ, nrZ.
Then the [n+ l, n]-rule with help of four actions of arithmetic and well known formulas
k=n∑
k=1
k = (1/2)n(n+ 1),
k=n∑
k=1
k2 = (1/6)n(n+ 1)(2n+ 1),
leads to the Klechkovski–Hakala formulas [11, 15]
Zn+l = K(n+ l) + 1, n+lZ = K(n+ l + 1),
Zn,l = K(n+ l + 1)− 2(l + 1)2 + 1, n,lZ = K(n+ l + 1)− 2l2,
Zn = K(n+ 1)− 1,
nZ = K(2n)− 2(n− 1)2 = (1/6)
[
(2n− 1)3 + 11(2n− 1)
]
,
Zl = K(2l + 1) + 1 = (1/6)
[
(2l + 1)3 + (5− 2l)
]
,
Znr = K(nr + 2)− 1. (2.1)
2See https://en.wikipedia.org/wiki/Aufbau_principle.
https://en.wikipedia.org/wiki/Aufbau_principle
Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 3
Here K(x) is the following function
K(x) = (1/6)x
[
x2 + 2− 3µ(x)
]
, x ∈ N,
µ(x) = x mod (2) =
{
1, x is odd,
0, x is even.
Relations for Zn+l and n+lZ are exact. Other relations are exact up to exclusions pointed
out above. For example, for l = 0, 1, . . ., we have Zl = 1, 5, 21, 57, . . ., that is, according to the
Madelung rule, the onset of the 4f block starts with La (Z = 57). But, according to the IUPAC
data3 on electron configurations in atoms, the onset of the 4f block starts with Ce (Z = 58).
The [n+ l, l]-rule defines essential characteristics of the Mendeleev periodic system.
In the Mendeleev periodic system, the table rows are enumerated by the Mendeleev number M
which is a linear function of the Madelung number M [15]:
M = M − 1 + δl,0 = n+ l − 1 + δl,0,
where δj,k is the Kronecker delta.
It is easy to show that the number of elements in the Mendeleev M-th period of the periodic
system according to the [n+ l, n]-rule is equal to
LM = K(M + 2)−K(M + 1) = 2
([
M
2
]
+ 1
)2
,
where [x] is an integer part of the real number x. Numbers LM, M = 1, 2, . . . form the sequence
2, 8, 8, 18, 18, 32, 32, . . . , which coincides with empirical lengths (1.1) of the periods of the system
of elements.
According to the [n+ l, n]-rule, the number ZM for the initial element of the Mendeleev M-th
period is equal to
ZM = K(M + 1)− 1,
and the number MZ for the final element of the Mendeleev M-th period is equal to
MZ = K(M + 2)− 2.
The sequence ZM = 1, 3, 11, 19, 37, 55, 87, . . . corresponds to alkaline metals, and the sequence
MZ = 2, 10, 18, 36, 54, 86, . . . corresponds to noble gases.
Thus, the empirical [n + l, n]-rule is a very efficient method to explain periodic table and
atomic properties. Research to justify it continues up to now (see, e.g., [2, 13, 14, 26, 27]. But
after 80 years of studies we have not yet good understanding for it. Standard textbooks on
quantum mechanics even do not mention this rule.
In the present paper we give a theoretical basis for the [n+ l, n]-rule. A preliminary version
of this paper was published in 2002 [6].
Further we use everywhere the atomic units ~ = e = m = 1, where ~ is the reduced Plank
constant, −e is the electron charge, and m is the electron mass.
3See https://iupac.org/what-we-do/periodic-table-of-elements/.
https://iupac.org/what-we-do/periodic-table-of-elements/
4 E.D. Belokolos
3 Atomic potential in the mean-field
and semi-classical approximations
The Hamiltonian of a free neutral atom is
HA =
Z∑
k=1
(
−1
2
∆k −
Z
~rk
)
+
1
2
Z∑
i,k=1
1
~rik
=
Z∑
k=1
(
−1
2
∆k + v(~rk)
)
,
v(~rk) = −Z
~rk
+
1
2
Z∑
i=1
1
~rik
,
where v(~r) describes the electron interactions with the atomic nucleus and other electrons.
Since it is good approximation to describe the electrons in an atom in terms of the electron
configuration, i.e., the electron distribution on one-particle states,
n, l, m, σ,
therefore we may consider the electron interaction v(~r) in the mean-field approximation with
the Hamiltonian
H =
Z∑
k=1
(
−1
2
∆k + V (~rk)
)
,
where V (~r) is the mean-field atomic potential. Since we consider a one-particle angular momen-
tum, quantum number l as a good quantum number, this potential has to be central,
V (~r) = V (r).
For the Schrödinger equation
HΨ = EΨ
we look for the ground state solution in the Slater determinant form
Ψ = det ||ψj(rk)||,
where one-particle wave function ψk(r) satisfies the one-particle Schrödinger equation describing
an electron in a 3-dimensional central potential V (r),(
−1
2
∆ + V (r)
)
ψj(r) = Ejψj(r).
In the semiclassical approximation, solutions of this equation have the form
ψ(r) = exp(iS(r)),
where S(r) is the classical action.
3.1 The Thomas–Fermi atomic potential
Since in the case under consideration we have three integrals of motion (energy E, angular
momentum L, and its projection Lz) our problem is integrable by quadratures due to the
Arnold–Liouville theorem, and the radial action looks as follows
S(r) =
1
π
∫ r [
2(E − V (r))−
I2θ
r2
]1/2
dr,
Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 5
where the radial action variable is
Ir =
1
π
∫ rmax
rmin
[
2(E − V (r))−
I2θ
r2
]1/2
dr.
In the semiclassical approximation we should change action variables by quantum numbers,
Ir = nr(E, l), Iθ = l.
In this way we come to the Bohr–Sommerfeld quantization rule
nr(E, l) =
1
π
∫ r+(E,l)
r−(E,l)
[
2E − 2V (r)− l2
r2
]1/2
dr.
Let us set in the latter expression E = 0, then
nr(0, l) =
√
2
π
∫ r+(0,l)
r−(0,l)
[
−V (r)− l2
2r2
]1/2
dr.
Thus the number of bound states N in the atom is
N =
lmax∑
l=0
nr(0, l)2(2l + 1).
Approximately a value N looks as follows
N '
∫ lmax
l=0
nr(0, l)2(2l + 1)dl
=
√
2
π
∫ r+
r−
∫ lmax
l=0
[
−V (r)− l2
2r2
]1/2
2(2l + 1)dldr
=
√
2
π
2
3
∫ r+
r−
[−V (r)]3/2 4r2dr =
2
√
2
3π2
∫ r+
r−
[−V (r)]3/2 4πr2dr =
∫ r+
r−
ρ(r)4πr2dr.
This is the well-known asymptotic formula [29] for the number of bound states in a central
potential with the density of bound states ρ(r) equal to
ρ(r) =
2
√
2
3π2
[−V (r)]3/2.
The electrostatic potential of atomic electrons φ(r) = −V (r) satisfies the Poisson equation
∆φ(r) = −4πρ, and therefore
∆φ(r) =
8
√
2
3π
φ3/2(r),
φ(r)r = Z, r → 0, φ(r) = 0, r →∞.
We can present the Thomas–Fermi potential φ(r) in such a way
φ(r) =
Z
r
χ(x), x =
r
R0
, R0 = bZ−1/3, b =
1
2
(
3π
4
)2/3
' 0.885.
For large Z the ground state energy ETF(Z) of the Thomas–Fermi atom is the asymptotics
for a ground state energy EHF(Z) of the Hartree–Fock atom [20, 21]:
lim
Z→∞
EHF(Z)/ETF(Z) = 1.
6 E.D. Belokolos
The Thomas–Fermi potential does not depend on quantum numbers although naturally it
should do. Nevertheless, let us calculate the Zl for the Thomas–Fermi potential [8].
The effective potential ul(r) is
ul(r) = −φ(r) +
(l + 1/2)2
2r2
= −(l + 1/2)2
2r2
[ζlxχ(x)− 1],
where
ζl =
2ZR0
(l + 1/2)2
= 2b
Z2/3
(l + 1/2)2
.
Conditions for the first appearance of the energy level with certain l are
ul(r) = 0, u′l(r) = 0
or, which is the same,
ul(x) = 0, u′l(x) = 0.
It is equivalent to equations
ζlxχ(x)− 1 = 0,
d
dx
(xχ(x)) = 0,
which have a solution
x0 ' 2.104, χ(x0) ' 0.231, x0χ(x0) ' 0.486,
χ′(x0) ' −0.110, ζl = [x0χ(x0)]
−1 ' 2.056.
This means that
ZTF
l
(2l + 1)3
=
(
ζl
8b
)3/2
= 0.155,
and hence
ZTF
l ' 0.155(2l + 1)3.
In the general case ZTFl is not integer and we have to write
ZTF
l =
∣∣0.155(2l + 1)3
∣∣,
where |x| means the integer which is the closest to the real x.
If we change in this formula the coefficient 0.155 to 0.169 ' 1/6 we get a better agreement of
the values ZTFl with that in the Mendeleev table (see [12] and [17, Section 73]). In this case the
r.h.s. of the formula will coincide with the first summand of the Klechkovski–Hakala expression
for Zl (2.1) obtained with the [n+ l]-rule.
This example shows that the Thomas–Fermi theory yields only some approximation to the
[n+ l, n]-rule.
According to the above discussion the electrons in the atom in the mean-field approximation
interact by means of central atomic potential, having geometrical O(3) symmetry. The Thomas–
Fermi theory takes into account the O(3) symmetry of the atomic Hamiltonian with central
potential. We do the next step. The Hamiltonian with any central potential has dynamical O(4)
symmetry [4, 10, 25], similar to the Hydrogen atom [5, 9, 28]. Therefore, the atomic Hamiltonian
must have an additional integral of motion in involution with respect to integrals of motion
corresponding to O(3) symmetry. Further we shall show that such additional integral of motion
in involution does exist for the atomic Hamiltonian and leads to [n+ l, n]-rule.
Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 7
3.2 The Tietz atomic potential
We begin with certain basic facts on symmetry and integrability in the classical Hamiltonian
systems (see [3, Sections 49–51], [16, Section 52] and [33]).
Let us assume that on a symplectic 2n-dimensional manifold there are n functions in involu-
tion
F1, . . . , Fn, {Fi, Fj} = 0, i, j = 1, . . . , n.
Consider a level set of functions Fi
Mf = {x : Fi(x) = fi, i = 1, . . . , n}.
Suppose that n functions Fi are independent on Mf (i.e., the n 1-forms dFi are linearly inde-
pendent in every point of Mf ).
Then
1. Mf is a smooth manifold, invariant with respect to the phase flow with the Hamiltonian
H = F1.
2. If the manifold Mf is compact and connected, then it is diffeomorphic to n-dimensional
torus
Tn = {(φ1, . . . , φn) (mod 2π)}.
3. Phase flow with the Hamiltonian H defines on Mf a quasi-periodic movement with angular
variables (φ1, . . . , φn),
dφi
dt
= ωi(f), φi(t) = φi(0) + ωit, i = 1, . . . , n.
Instead of functions F = (F1, . . . , Fn) it is possible to define new functions I = (I1(F ), . . . ,
In(F )) which are called action variables and which together with angle variables form in
the neighborhood of manifold Mf the canonical system of action-angle coordinates.
4. Canonical equations with the Hamiltonian function are integrable in quadratures.
5. If frequencies ω = (ω1, . . . , ωn) are degenerated, i.e., if there exists such an integer-valued
vector k = (k1, . . . , kn) ∈ Zn that
(k, ω) = 0,
then there appears one more single-valued function Fn+1 which is in involution with func-
tions F1, . . . , Fn.
Now let us go back to our problem: an electron in the central atomic potential. In this case
the 3-dimensional electron movement is reduced to 2-dimensional one in the plane perpendicular
to the orbital momentum vector. We describe this movement in the semiclassical approximation
by the action
S =
∫ r
√
2[E − V (r)]−
I2θ
r2
dr
with radial (r, Ir) and orbital (θ, Iθ) action-angle variables, where
Ir =
1
π
∫ rmax
rmin
√
2[E − V (r)]−
I2θ
r2
dr.
8 E.D. Belokolos
And so in our problem we have a quasi-periodic movement on 2-dimensional torus which is
described by the Fourier series
G(t) =
∑
l1∈Z
∑
l2∈Z
Gl1,l2 exp(l1φr + l2φθ) =
∑
l1∈Z
∑
l2∈Z
Gl1,l2 exp[(l1ωr + l2ωθ)t]
with 2 basic frequencies
ωr =
∂E
∂Ir
, ωθ =
∂E
∂Iθ
.
In the general case these frequencies are independent. But under certain circumstances, as we
have, when new additional integral appears, they may become degenerate (or commensurate, or
resonance),
qωr = pωθ, q, p ∈ Z, g.c.d.(p, q) = 1,
where g.c.d.(p, q) means the greatest common divisor of the integers p and q. This condition
leads to important consequences.
1. Since the latter relation is equivalent to
q
∂E
∂Ir
= p
∂E
∂Iθ
it means that energy E depends on pIr + qIθ:
E = E(pIr + qIθ).
And due to the Bohr–Sommerfeld semiclassical quantization rule
Ir → nr, Iθ → l,
we have
E = E(pnr + ql).
2. We have the additional independent integral of motion.
3. The canonical variables of the problem are separated in several systems of coordinates.
For example, for the Kepler–Coulomb potential 1/r we have p = q = 1, energy E depends on
principal quantum number n = nr + l+ 1, the canonical variables of the problem are separated
in polar and parabolic coordinates.
Let us study the similar situation for the atomic potential. Consider a system of equations
Ir(E) =
1
π
∫ r+(E)
r−(E)
[
2E − 2V (r)−
I2θ
r2
]1/2
dr,
Mp,q = pIr + qIθ, q, p ∈ Z, g.c.d.(p, q) = 1,
the first of which is the standard relation between action variables Ir, Iθ, and energy E in
central potential and the second one is relation between Ir, Iθ arising as result of frequencies
degeneracy.
Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 9
We shall study equations at the energy E = 0 at which new bound states appear from
continuous spectrum, when Z grows. Then differentiating equations with respect to Iθ we come
to the equation
πα =
∫ r+
r−
[
− 2V (r)r2 − I2θ
]−1/2
Iθ
dr
r
,
where
α = −∂Ir
∂Iθ
=
q
p
∈ Q, q, p ∈ Z, g.c.d.(p, q) = 1.
From this equation we deduce an expression for atomic potential V (r). A similar integral
equation arising in problem for generalized tautochrone (isochrone) curve was solved by Abel [1].
We shall use his method in the form presented in [16, Section 12].
Theorem 1. Equation
πα =
∫ r+
r−
[
−2V (r)r2 − L2
]−1/2
L
dr
r
(3.1)
has a solution
Vα,β,R(r) = − β
r2[(r/R)1/α + (R/r)1/α]2
,
where β and R are certain constants, and its deformations.
If the potential Vα,β,R(r) coincides at small r with the Kepler–Coulomb potential
Vα,β,R = −Z/r, r → 0,
then we have
α = q/p = 2, β = ZR,
and the potential takes the following form
V (r) ≡ V2,ZR,R(r) = − Z
r(1 + (r/R))2
.
Remark 1. According to the theorem in the neighborhood E ' 0 we have
E = E(n+ l)
and this fact proves the first part of the [n + l, n]-rule. A second part of the [n + l, n]-rule is
a consequence of the oscillation theorem.
Proof. Let us rewrite equation (3.1) in the form
πα =
∫ r+
r−
[
w(x)− L2
]−1/2
Ldx,
where
x = ln(r/R), r = R exp(x),
w(x) = −2V (r)r2 at r = R exp(x),
w0 = maxw(x),
10 E.D. Belokolos
are new variables and R is a parameter. We assume that w(x) is the one-well potential and
therefore the inverse function is two-valued, i.e., the values w are reached in two points x−(w)
and x+(w). We shall assume also that x−(w) ≤ x+(w) and at w0 we have x−(w0) = x+(w0).
As a result we obtain
πα =
∫ w0
L2
(
w − L2
)−1/2(dx+
dw
− dx−
dw
)
Ldw.
Multiplying this equality by
(
L2 − w1
)−1/2
2dL and integrating from (w1)
1/2 to (w0)
1/2 we
get
2πα
∫ (w0)1/2
(w1)1/2
(
L2 − w1
)−1/2
dL
=
∫ (w0)1/2
(w1)1/2
2LdL
∫ w0
L2
(
dx+
dw
− dx−
dw
)[(
w − L2
)(
L2 − w1
)]−1/2
dw
=
∫ w0
w1
dw
(
dx+
dw
− dx−
dw
)∫ (w)1/2
(w1)1/2
[(
w2 − L2
)(
L2 − w1
)]−1/2
2LdL.
Since ∫ (w0)1/2
(w1)1/2
(
L2 − w1
)−1/2
dL = Arcosh(w0/w1)
1/2,∫ (w)1/2
(w1)1/2
[(
w2 − L2
)(
L2 − w1
)]−1/2
2LdL = π,
this equality acquires the following form
2αArcosh(w0/w1)
1/2 = (x+(w)− x−(w))
∣∣∣w0
w1
.
Taking into account that x−(w0) = x+(w0) and setting w1 = w we come to the equality
x+(w)− x−(w) = 2αArcosh(w0/w)1/2.
This equality defines only the difference x−(w)− x+(w) of two functions x−(w) and x+(w),
any of which remains actually undefined. It means that there exists the infinite set of potentials
which satisfy the latter equation and differ by deformations which do not change the difference
of two values of x corresponding to one value of w. Among these potentials there is a symmetric
potential with the property x+(w) = −x−(w) ≡ x(w). For this potential in this case
w(x) = w0/ cosh2(x/α),
or, in the previous notations,
Vα,β,R(r) = − β
r2[(r/R)1/α + (R/r)1/α]2
,
where β = 2w0. These potentials are degenerate for arbitrary value of the parameter α. In order
for the degenerate potential to coincide with Coulomb potential at small r
Vα,β,R = −Z/r, r → 0,
we must have
α = q/p = 2, β = ZR.
Then this potential takes the following form
V (r) ≡ V2,ZR,R(r) = − Z
r(1 + (r/R))2
. �
Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 11
In this potential the total number of bound states N is equal to the total number of elect-
rons Z if
N =
8
√
2
3π
∫
[−V (r)]3/2dr =
(
9/2R3
)
= Z,
i.e.,
R = (9/2Z)1/3 ' 1.651Z−1/3,
and we have finally
V (r|Z) = − Z
r(1 + (r/R(Z)))2
.
For the given electron configuration we must use the Klechkovski–Hakala formulas for Z. In
this case the potential will depend on quantum numbers.
We can present this potential as the sum
V (x) = − (Z/R)
x(1 + x)2
= −Z
R
[
1
x
− 1
(x+ 1)
− 1
(x+ 1)2
]
, x =
r
R
,
where the first summand describes the Coulomb attraction of the atomic nucleus for the single
electron and two other summands describe a nucleus screening by the other electrons.
T. Tietz [30] proposed the potential
V (r) = − Z
r[1 + (r/R)]2
as a good rational approximation to the Thomas–Fermi potential and used it for calculation of
various atomic properties and explanation of the periodic system elements [31] (see also [34]).
Further we shall call this potential as the Tietz atomic potential.
Due to the proximity of the Tietz and Thomas–Fermi potentials it is very likely that at
large Z a ground state energy of the Tietz atom, ET(Z), is an asymptotics for the ground state
energy of the Hartree–Fock atom, EHF(Z):
lim
Z→∞
EHF(Z)/ET(Z) = 1.
Yu.N. Demkov and V.N. Ostrovski [7] pointed out that this potential is a particular case of
a so called focussing (in other words “degenerate”) potentials studied in connection to certain
problems of optics by J.C. Maxwell [23] and V. Lenz [19].
The Tietz potential is a rational function, and thus we can easily do various calculations with
it. For example, we can calculate the atomic spectrum.
4 Semiclassical atomic spectrum for the Tietz atomic potential
The Bohr–Sommerfeld semiclassical condition of quantization for the Tietz atomic potential is
nr =
1
π
∫ r+
r−
[
2E +
2Z
r(1 + (r/R))2
− (l + 1/2)2
r2
]1/2
dr.
In the scaled quantum numbers
x =
r
R
, ε =
2ER2
(l + 1/2)2
, νr =
nr
l + 1/2
, ηl =
2ZR
(l + 1/2)2
12 E.D. Belokolos
the Bohr–Sommerfeld equation takes the form
νr =
1
π
∫ x+
x−
[
ε+
ηl
x(1 + x)2
− 1
x2
]1/2
dx =
1
π
∫ x+
x−
√
P4(x)
x(1 + x)
dx,
where P4(x) is a 4-th degree polynomial
P4(x) = εx2(1 + x)2 + ηlx− (1 + x)2,
and boundaries of integration are real non-negative zeros of this polynomial. Therefore scaled
radial quantum number νr is a period of the elliptic integral with scaled energy ε and scaled
charge ηl as the parameters. We can obtain an atomic spectrum in the semiclassical approxi-
mation
ε = f(νr, ηl),
by means of inversion of the elliptic integral.
Let us study this problem in the framework of perturbation theory in the energy ε. We
present the Bohr–Sommerfeld semiclassical equation in the form
νr =
1
π
∫ x+
x−
[
ε+
ηlx− (1 + x)2
x2(1 + x)2
]1/2
dx
=
1
π
∫ x+
x−
[
ε+
(x+ − x)(x− x−)
x2(1 + x)2
]1/2
dx,
where
ηlx− (1 + x)2 = (x+ − x)(x− x−),
x2± − (ηl − 2)x± + 1 = 0,
x± =
(ηl − 2)±
√
(ηl − 2)2 − 4
2
=
(ηl − 2)±
√
ηl(ηl − 4)
2
,
−(ηl − 4) > ε > −1
4
ηl(ηl − 4).
At the first approximation we have
νr ' J0 + εJ1
=
1
π
∫ x+
x−
[(x+ − x)(x− x−)]1/2
dx
x(1 + x)
+ ε
1
2π
∫ x+
x−
x(1 + x)
[(x+ − x)(x− x−)]1/2
dx,
where
J0 =
1
π
∫ x+
x−
[(x+ − x)(x− x−)]1/2
dx
x(1 + x)
=
√
ηl − 2,
J1 =
1
2π
∫ x+
x−
x(1 + x)
[(x+ − x)(x− x−)]1/2
dx =
1
16
(
3η2l − 8ηl
)
.
The energy spectrum in the semiclassical theory is
εn,l = −J0 − νr
J1
= −16
√
ηl − 2− νr(
3η2l − 8ηl
) = −16
(l + (1/2))
√
ηl − 2l − 1− nr
(l + (1/2))
(
3η2l − 8ηl
)
= −16
(l + (1/2))
√
ηl − (n+ l)
(l + (1/2))
(
3η2l − 8ηl
) = −16
√
2ZR−M
(l + (1/2))
(
3η2l − 8ηl
) ,
where we have used the Madelung number M = n+ l.
Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential 13
Since
εM,l =
2EM,lR
2
(l + (1/2))2
,
we have the following expression for the atomic spectrum in the atomic units
EM,l = −8
(
√
2ZR−M)(l + (1/2))(
3η2l − 8ηl
)
R2
, ηl =
2ZR
(l + (1/2))2
.
If
√
2ZR = M
then EM,l = 0, and all states (M, l) are degenerate with respect to l. Thus at energy E = 0 there
appears the full set of energy levels (n, l) with M = n + l. Since R = (9/2Z)1/3 the equality
M =
√
2ZR is equivalent to
Z = (1/6)M3.
We can write down a complete perturbation series
νr =
∞∑
k=0
εkJk. (4.1)
Since all integrals
Jk =
ck
π
∫ x+
x−
{
x2(1 + x)2
[(x+ − x)(x− x−)]
}k−(1/2)
dx,
ck =
Γ(3/2)
Γ(k + 1)Γ((3/2)− k)
, k ≥ 2,
are divergent we should take their regularized values. i.e., valeur principale. Inverting series (4.1)
by means of the Bürmann–Lagrange theorem we can obtain an expression for atomic spectrum
in the semiclassical approximation
ε = f(νr, ηl).
We plans to compare this atomic spectrum (both eigenvalues and eigenfunctions) with those
presented in articles [18, 32].
5 Conclusion
For the Mendeleev periodic system of elements we have proved the empirical (n+l, n)-rule which
explains very efficiently the structure and properties of chemical elements. In order to prove
this rule we are forced to build for the Hamiltonian describing an electron in central atomic
potential one more integral of motion in involution in addition to energy and orbital integral.
In this case the atomic potential appears to be the Tietz potential. For the Tietz potential we
have calculated the atomic energy spectrum in the semiclassical approximation.
In addition, we are going to study spectrum of the Schrödinger operator with the Tietz
potential, when the radial part of wave function satisfies the confluent Heun equation and
compare energy levels, obtained in such a way, with the NIST Atomic Spectra Database.
14 E.D. Belokolos
For the atom in the paper we have studied the non-relativistic atomic Hamiltonian. We plan
to consider also the relativistic Dirac Hamiltonian.
It is interesting also to study isospectral many-well deformation of the Tietz potential.
Questions studied in the paper are related only to the ground state of atoms. But according
to experimental data the weakly excited atomic states also follow to [n+ l, n]-rule [15]. It may
be of interest for physical chemistry, especially for understanding chemical reactions rules.
We hope to do that in the near future.
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1 Introduction
2 The Madelung [n+l,n]-rule
3 Atomic potential in the mean-field and semi-classical approximations
3.1 The Thomas–Fermi atomic potential
3.2 The Tietz atomic potential
4 Semiclassical atomic spectrum for the Tietz atomic potential
5 Conclusion
References
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